Joseph Kreimer

Sensitivity analysis and the “what if” problem in simulation analysis

We discuss some known and some new results on the score function (SF) approach for simulation analysis. We show that while simulating a single sample path from the underlying system or from an associated system and applying the Radon-Nikodym measure one can: estimate the performance sensitivities (gradient, Hessian etc.) of the underlying system with respect to some parameter (vector of parameters); extrapolate the performance measure for different values of the parameters; evaluate the performance measures of queuing models working in heavy traffic by simulating an associated (auxiliary) queuing model working in light (lighter) traffic; evaluate the performance measures of stochastic models while simulating random vectors (say, by the inverse transform method) from an auxiliary probability density function rather than from the original one (say by the acceptance-rejection method). Applications of the SF approach to a broad variety of stochastic models are given.

Nondifferentiable optimization via smooth approximation: general analytical approach

In this paper we present a method for nondifferentiable optimization, based on smoothed functionals which preserve such useful properties of the original function as convexity and continuous differentiability. We show that smoothed functionals are convenient for implementation on computers. We also show how some earlier results in nondifferentiable optimization based on smoothing-out of kink points can be fitted into the framework of smoothed functionals. We obtain polynomial approximations of any order from smoothed functionals with kernels given by Beta distributions. Applications of smoothed functionals to optimization of min-max and other problems are also discussed.

An optimal operation policy for real-time n-server stand-by systems involving preventive maintenance

Abstract An optimal operation policy for n -server stand-by systems operating in real time is obtained. It appears that, under certain conditions, such a policy is achieved by successive total use of each server capacity. The analysis is performed in two stages: (1) a non-stop operating system is treated, and then, (2) a general system, which operates only in response to orders. As demonstrated, such a consecutive approach seems to be highly promising for the analysis of complicated systems involving preventive maintenance and operating in real time.

Optimal Real-Time Data Acquisition and Processing by a Multiserver Stand-by System

We obtain an optimal operation policy for a general n-server channel, machine stand-by system involving preventive maintenance and operating in real time. We show that such a policy is achieved by successive total use of each server capacity, and does not depend on the arrival pattern of processed data and task duration. We show also that this policy is optimal for any finite, initial period of time, and not only in the sense of long-run availability. The analysis is performed in two stages: a worst-case analysis is performed, and then, a general case is treated. Such a consecutive approach also seems to be useful in the analysis of more complicated systems. Rigorous mathematical proofs are provided. Actual and possible applications of results to military and production control systems are discussed.

An optimal routing policy for unmanned aerial vehicles (analytical and cross-entropy simulation approach)

We consider a real-world problem of military intelligence unit equipped with identical unmanned aerial vehicles producing real-time imagery and responsible for heterogeneous regions (with requests of real-time jobs) required to be under nonstop surveillance. Under certain assumptions these real-time systems can be treated as queueing networks.The use of the system involving unmanned aerial vehicles relies on the principle of availability, namely on its ability to process the maximal portion of real-time tasks. We show that even very large number of vehicles does not guarantee the maximal system availability without proper choice of routing probabilities. We compute analytically (for exponentially distributed service and maintenance times) and via simulation using Cross-Entropy method (for generally distributed service times) optimal routing probabilities which maximize system availability.

Effectiveness-analysis of real-time data acquisition and processing multichannel systems

Consider a real-time data acquisition and processing multiserver (e.g., unmanned air vehicles and machine controllers) and multichannel (e.g., surveillance regions, communication channels, and assembly lines) system involving maintenance. In this kind of system, jobs are executed immediately upon arrival, conditional on system availability. That part of the job which is not served immediately is lost forever and cannot be processed later. Thus, queuing of jobs in such systems is impossible. The effectiveness analysis of real-time systems in a multichannel environment is important. Several definitions of performance effectiveness index for the real-time system under consideration are suggested. The real-time system is treated with exponentially distributed time-to-failure, maintenance, interarrival and duration times as a Markov chain in order to compute its steady-state probabilities and performance effectiveness index via analytic and numerical methods. Some interesting analytic results are presented concerning a worst-case analysis, which is most typical in high-performance data acquisition and control real-time systems.

Real-time system with homogeneous servers and nonidentical channels in steady-state

A real-time multiserver system with homogeneous servers (such as unmanned air vehicles or machine controllers) and several nonidentical channels (such as surveillance regions or assembly lines) working under maximum load regime is studied. We show how to compute steady-state probabilities of such a system, when both maintenance and service times are exponentially distributed. We also compute various important performance parameters, including system availability and loss penalty function.

Real-time multiserver and multichannel systems with shortage of maintenance crews

We consider a real-time multiserver (such as unmanned air vehicles or machine controllers) and multichannel (such as surveillance regions or assembly lines) systems working under maximum load regime with shortage of maintenance crews (less than the number of servers). In such systems, a job arriving is served immediately upon arrival (conditional on system availability) without delay. That part of the job which is not served immediately is lost forever and cannot be processed later. We show how to compute equilibrium (steady state) probabilities of real-time systems, when both maintenance and service times are exponentially distributed (birth and death stochastic process). Then we also compute various important system parameters (including its availability) and provide some numerical results.

Optimization of real-time multiserver system with two different channels and shortage of maintenance facilities

We present optimality conditions for real-time multiserver system with large number of identical servers (e.g. unmanned air vehicles, machine controllers, etc.) and two non-identical channels (e.g. surveillance regions, assembly lines, etc.) working under maximum load regime with limited maintenance facilities. Real-time systems are responsible for operations management of increasingly sensitive applications, particularly those in which failures to satisfy temporal restrictions can lead to serious or even dangerous consequences. Optimization of these systems is very important. We calculate limiting values of system availability and its loss penalty function and show how to obtain optimal assignment probabilities which optimize (maximize and minimize, respectively) these Performance measures.

Real-time multiserver with two non-identical channels and limited maintenance facilities

A real-time multiserver (such as unmanned air vehicles or machine controllers) system with two different channels (such as surveillance regions or assembly lines) working under maximum load regime with limited maintenance facilities is studied. In such system, a job is served immediately upon arrival (conditional on system availability) without delay. That part of the job which is not processed immediately is lost forever and cannot be served later. We show how to compute steady-state probabilities of such a system, when both maintenance and service times are exponentially distributed. Then, we also compute various important performance characteristics, including system availability and loss penalty function.